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It is NOT known if Archimedes suspected that there was no exact answer for numbers like √2, √3, √5 and π. The proof that pi is transcendental would have to wait for the esteemed German mathematician Carl Louis Ferdinand von Lindemann (1852-1939) in 1882. This means, by the way, that the ancient problem of squaring the circle - constructing a square equal in area to a given circle using only compasses and a straight edge - had no solution. However, one can come very close to an exact construction. Archimedes himself approximated pi as between 22/7 and 223/71, or ~3.142857 and ~ 3.1408 with a real value (to 8 places) of ~3.14159265. At this writing pi had been calculated to 1.2 TRILLION
places in 2002.
UPDATE: Emma Haruka Iwao, a Google employee, broke the Guinness world record for calculating the most digits of Pi. Iwao and a Google team computed Pi to 31.4 trillion decimal places or Pi multiplied by 10 to the 13th power, ousting the previous record set in 2016 of 22.4 trillion digits. Of interest is how Archimedes obtained his results. Many mathematicians, notably the formidable John Wallis, found it difficult to understand how Archimedes worked. For centuries sand was useful for beaches and eventually glass, but it took a very long time before several geniuses worked out how to make a glass lens out of sand and provided us with telescopes and microscopes. Fifty years ago some even more modern transformations of sand grains resulted in devices neither Wallis nor Archimedes could have ever imagined.
In another work, Measurement of a Circle, Archimedes pointed out
the square root of three (√3) lies between 265/153 and 1351/780.
Archimedes could only work in fractions - taking advantage of
contemporary decimal notation we would say ~1.7320261 and
~1.7320512. The actual value is close to 1,7320508. In the same work Archimedes gave an algorithm for determining the ratio of a circle's diameter to its area (usually abbreviated as pi or from the Greek π) to an arbitrary precision.